Problem 24 - Precalculus Mathematics for Calculus

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. $$\frac{2 x-1}{x+2}=\frac{4}{5}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{13}{6}\).

Step 1: Cross Multiply

To clear the fractions by cross-multiplying, multiply both sides of the equation by the denominators. We obtain:\[(2x - 1) imes 5 = 4 imes (x + 2)\]Leading to the equation:\[10x - 5 = 4x + 8\]

Step 2: Move All Terms Involving x to One Side

To isolate the variable, move all terms involving \(x\) to one side of the equation:\[10x - 4x = 8 + 5\]Thereby simplifying to:\[6x = 13\]

Step 3: Solve for x

To find the value of \(x\), divide both sides by the coefficient of \(x\) which is 6:\[x = \frac{13}{6}\]

Key Concepts

Cross-multiplication

Cross-multiplication is a powerful method used to solve equations involving fractions. Imagine having fractions on both sides of an equation. This can seem a bit tricky at first! But, by using cross-multiplication, you can simplify the process.To cross-multiply, first identify the denominators of the fractions. In our given problem, we have two fractions: \( \frac{2x-1}{x+2} \) and \( \frac{4}{5} \). Here's what you do:

• Multiply the numerator (the top part) of the first fraction by the denominator (the bottom part) of the second fraction.
• Do the same with the other fraction: multiply the numerator of the second by the denominator of the first.

This gives us the equation \((2x - 1) \times 5 = 4 \times (x + 2)\).See how it clears those pesky fractions? Now, we're dealing with a straightforward equation that’s easier to handle.

Solve for x

Solving for \(x\) means finding the value of \(x\) that makes the equation true. After using cross-multiplication, you are usually left with a simpler equation.In our scenario, we are left with the equation \(10x - 5 = 4x + 8\). The next steps involve simplifying this equation to find the exact value of \(x\).

• First, simplify each side of the equation if needed. Here, we're already quite simplified!
• The equation is now ready for further simplification: combine like terms. This means we've got to get all those \(x\)'s on one side.

The goal is to make \(x\) "stand alone." In this step, once terms involving \(x\) are together, you simplify further by adding or subtracting constants from both sides.

Isolate variable

Isolating a variable like \(x\) in an equation means rearranging the equation so that \(x\) is by itself on one side. This often involves moving terms around and using basic arithmetic operations.In our working equation \(10x - 4x = 8 + 5\), we start by getting terms with \(x\) on one side and constants on the other:

• Subtract or add terms on both sides to simplify. For instance, subtract \(4x\) from both sides in this case.
• This leads to \(6x = 13\). Now, \(x\) is multiplied by 6.

To isolate \(x\), divide both sides by the coefficient of \(x\), which is 6. This final step leaves us with our solution: \(x = \frac{13}{6}\).By isolating \(x\), we have found the specific value of \(x\) that satisfies the original equation. Isn’t that satisfying? :)